There are many reasons this happened. It was different in case of Mauryas and the Mughals.
- At the time of Mauryas, State formation in Deccan and South India was still in its formative stages. Most of the conquests of the Mauryan Empire conlcuded with Bindusara. Ashoka could have very easily conquered South India( including Tamil Nadu) had he wanted, because it would have been a superior organisational structure meeting an inferior one. But then what? The difficulty in administering such a distant land from the capital would have rendered such a mission useless.
- During the Mughal Period, again distance was the main factor. Here there was another factor too. State formation was complete by this time and Mughals would have had to contend with powerful regional states and this would have meant an enormous drain on the resources of the empire. The last powerful Mughal, Aurangzeb, could not even properly subdue the powerful Marathas. What would the Mughals do with an even distant Tamil Nadu. The much easier option was what Delhi Sultans took. Plunder, and accept tribute and retreat.
- Ashoka, after the Kalinga war took to a policy of “Dhammavijaya”, which is what makes him the most distinguished ruler in the ancient world. Some historians describe him as “Monk in a monarch’s garb”. After Kalinga, which is the only territory Ashoka annexed, he adopted a friendly policy to the peripheries. Therefore his edicts mentions Satiyaputras and others as friendly neighbours.
- Both empires were highly centralised in character. In a land like India, such centralisation is counterpoised by regional tendencies. This meant that both empires had their hands full trying to deal with the regions they had already captured. Consolidation of these parts itself kept them occupied from not venturing further south.
There are many more complicated reasons which would require me to delve into the details of administration and polity of these two Empires. However, the above points cover the gist of it.
Hope this helps.